3. (40 points) Let us consider the Limit Design of the 4-bar elastic-perfectly plastic truss shown...

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3. (40 points) Let us consider the Limit Design of the 4-bar elastic-perfectly plastic truss shown below. Assume that all the given numeric problem data are in consistent units. E, A same for all elements 0.5 m 0.5 m 1.0 m 2 12 0.3 m x 0.7 m 5 We need to obtain an optimum design a* = (a,a,a,a) for this truss that maximizes the limit load factor & such that the stresses in members are within allowable limits, and that the total volume does not exceed a given value. The nominal forces acting on node 2, of magnitude 12 and 8 respectively, are shown in the figure. The allowable yield stress limits in tension and compression are +9 and -9, respectively, for each of the bars, and the bound on the total volume V is 250. 1. State a formal design optimization problem that, when solved, will provide the optimum de- sign that we seek. 2. Recall that optimization problem: minimize obj cx such that A x b Aeq. x = beq x lb x ub can be solved by the MATLAB linprog( ) command as follows: [x_opt, obj_opt]=linprog (c, A, b, Aeq, beq, 1b, ub) Write a complete MATLAB script that will compute the optimum limit design for the given 3-bar truss. cos e sin 20 -cos e sin 20 Ee Ae sin 20 sin e -sin 20 - sin e [K]= he - cos e 2 sin 20 cos Be sin 20 sin 20 - sin O sin 20 sin e 2 - (A) + 2(A)'. =1-3 = 3 2 -2 ( 4 ) - = x =-x * = (1 - 2 - he 2 [(A) - *] 6 -3he -6 -3he 2Eele -3he 2h2 3he 2h 3he h [K]= h -6 3he 6 3he -3he h 3he 2h2 6 Q1 gehe -he Q2 {Fe} = + 12 6 Q3 he Q4 W1 COS sina 0 - sin a cos a 0 WI S 001 S (5.4.6a) 2 W2 0 cos a - sin a sina 0 142 cosa 0 W 52 00 1 S 3. (40 points) Let us consider the Limit Design of the 4-bar elastic-perfectly plastic truss shown below. Assume that all the given numeric problem data are in consistent units. E, A same for all elements 0.5 m 0.5 m 1.0 m 2 12 0.3 m x 0.7 m 5 We need to obtain an optimum design a* = (a,a,a,a) for this truss that maximizes the limit load factor & such that the stresses in members are within allowable limits, and that the total volume does not exceed a given value. The nominal forces acting on node 2, of magnitude 12 and 8 respectively, are shown in the figure. The allowable yield stress limits in tension and compression are +9 and -9, respectively, for each of the bars, and the bound on the total volume V is 250. 1. State a formal design optimization problem that, when solved, will provide the optimum de- sign that we seek. 2. Recall that optimization problem: minimize obj cx such that A x b Aeq. x = beq x lb x ub can be solved by the MATLAB linprog( ) command as follows: [x_opt, obj_opt]=linprog (c, A, b, Aeq, beq, 1b, ub) Write a complete MATLAB script that will compute the optimum limit design for the given 3-bar truss. cos e sin 20 -cos e sin 20 Ee Ae sin 20 sin e -sin 20 - sin e [K]= he - cos e 2 sin 20 cos Be sin 20 sin 20 - sin O sin 20 sin e 2 - (A) + 2(A)'. =1-3 = 3 2 -2 ( 4 ) - = x =-x * = (1 - 2 - he 2 [(A) - *] 6 -3he -6 -3he 2Eele -3he 2h2 3he 2h 3he h [K]= h -6 3he 6 3he -3he h 3he 2h2 6 Q1 gehe -he Q2 {Fe} = + 12 6 Q3 he Q4 W1 COS sina 0 - sin a cos a 0 WI S 001 S (5.4.6a) 2 W2 0 cos a - sin a sina 0 142 cosa 0 W 52 00 1 S